Most of the following discussion should be familiar to you. I am writing it here for completeness. Also, it’s a good review for me.
Assume one spring/mass/damper suspension system is set up vertically under a car with one wheel. Assume also that the system is bolted to the car and the wheel stays on the road. The y position of the wheel at any point in time is the elevation of the road (which changes over time) plus the radius of the wheel (which is constant). The y position of the car is the y position of the wheel plus the equilibrium height of the suspension (constant) plus the deflection of the suspension (changes over time).
You already know that when you first place the car on the road, the spring/mass/damper combined with the weight of the car will cause the elevation of the car to oscillate in smaller and smaller increments until it finally comes to rest at an equilibrium height above the road. You should have the derivation for this “damped oscillation under a constant external force” in your textbook.
If the road is flat, the y position of the wheel is constant. The y position of the car is solved when you fill in the values for all the variables the above mentioned equation. But since the road has hills, there is an additional force on the spring/mass/damper and that force is not constant. As the car goes up and down the hills the external force is equal to the weight of the car plus the force necessary to elevate/lower the car to keep it on the road. You can model this by thinking of the weight of the car as varying. When the car goes up a hill, it gets heavier. When it goes down a hill, it gets lighter.
So now you need to ask “what is the mathematical function for the varying force/weight on/of the car?”. The teacher is making it easy for you by specifying that the hills are sinusoidal in shape. Sinusoidal shaped hills means the vertical deflection of the wheel varies by the sine, the vertical velocity varies by the cosine and the vertical acceleration varies by the sine. Force is mass times acceleration. So the external force on the wheel as a result of the hills is the mass of the car times the sine function. The total external force on the wheels (apparent weight) is the real weight of the car plus this sine function. You can model this total external force as a single sine function. When you do that you have a spring/mass/damper system under a sinusoidal forcing function. You should have a generic derivation for that arrangement in your textbook too.
That’s the setup. If you need help with the actual equations I can go through them with you too. But I should really wait till after the exam for that.
